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numthy · MillerFunction

11 accepted declarations. Python and Rust forms, contracts, evidence, failure boundaries, and checked usages are kept together below.

Worked example

The complete checked program Reproduce Fisher's order-five elliptic-function normalization uses this API directly.

# SPDX-License-Identifier: MIT
# Copyright (c) 2026 Benjamin Antieau and Deven Manam
"""Reproduce Fisher's order-five elliptic-function normalization."""

from noether import Q
from noether.commalg import RationalFunctionField, WeilDivisor
from noether.numthy import EllipticCurve, NthPowerClassGroup


def main() -> None:
    coefficients = RationalFunctionField(Q, "lambda")
    parameter = coefficients.generator()
    curve = EllipticCurve(
        coefficients,
        [1 - parameter, -parameter, -parameter, 0, 0],
    )
    point = curve.point(0, 0)
    point.verify_exact_order(5)

    miller = curve.miller_function(point, 5)
    divisor = WeilDivisor(
        curve.underlying_curve,
        [
            (curve.origin().underlying_curve_point.as_closed_point(), 5),
            (point.underlying_curve_point.as_closed_point(), -5),
        ],
    )
    function = divisor.riemann_roch_space().basis[0]
    assert (function / miller.function).as_constant() is not None

    evaluation_at_two = function.evaluate_at((2 * point).underlying_curve_point)
    evaluation_at_four = function.evaluate_at((4 * point).underlying_curve_point)
    assert evaluation_at_two.value is not None
    assert evaluation_at_four.value is not None
    classes = NthPowerClassGroup(coefficients, 5)
    normalization = classes.class_of(evaluation_at_two.value) ** -2 * classes.class_of(
        evaluation_at_four.value
    )
    assert normalization.is_identity()
    print("order 5 verified; Riemann--Roch and Miller agree; normalization is trivial")


if __name__ == "__main__":
    main()

Run it with .venv/bin/python examples/26_elliptic_descent_functions.py.

numthy.MillerFunction

A Miller function with its exact divisor formula.

class MillerFunction

noether::numthy::MillerFunction · Direct Rust declaration

pub struct MillerFunction<R: Field> { /* private fields */ }
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core
GIL attached / bounded
Semantic ID noether/numthy/MillerFunction@1

Executable usage: validation/unit/numthy/test_elliptic_curve_facade.py:11 · validation/unit/numthy/test_elliptic_curve_facade.py:98

numthy.MillerFunction.curve

Return the curve recorded by this MillerFunction.

Detailed contract

Owning elliptic curve.

(self) -> EllipticCurve[E]

noether::numthy::MillerFunction::curve · Direct Rust declaration

pub fn curve(&self) -> &EllipticCurve<R>
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core
GIL attached / bounded
Semantic ID noether/numthy/MillerFunction@1

Executable usage: examples/26_elliptic_descent_functions.py:13 · examples/26_elliptic_descent_functions.py:17 · examples/26_elliptic_descent_functions.py:20 · examples/26_elliptic_descent_functions.py:22 · examples/26_elliptic_descent_functions.py:24 · examples/37_complete_intersection_theory.py:27

numthy.MillerFunction.divisor

Return the divisor recorded by this MillerFunction.

(self) -> WeilDivisor[E]

noether::numthy::MillerFunction · Python projection

Python projection convenience over this Rust mathematical type
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core
GIL attached / bounded
Semantic ID noether/numthy/MillerFunction@1

Executable usage: examples/26_elliptic_descent_functions.py:21 · examples/26_elliptic_descent_functions.py:28 · validation/unit/commalg/test_curve_facade.py:114 · validation/unit/commalg/test_curve_facade.py:116 · validation/unit/commalg/test_curve_facade.py:122 · validation/unit/commalg/test_curve_facade.py:125

numthy.MillerFunction.from_json

Deserialize and revalidate a MillerFunction from typed JSON.

Detailed contract

Restore the source point and rerun Miller’s algorithm.

(cls, value: str) -> MillerFunction[EllipticBase]

noether::numthy::MillerFunction::from_json · Direct Rust declaration

pub fn from_json(value: &str) -> Result<Self, NoetherError>where R: DeserializeOwned, R::Element: DeserializeOwned,
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core
GIL attached / bounded
Semantic ID noether/numthy/MillerFunction@1

Executable usage: validation/unit/category/test_category.py:14 · validation/unit/combin/test_acceptance_workload.py:65 · validation/unit/combin/test_acceptance_workload.py:86 · validation/unit/combin/test_graph_facade.py:106 · validation/unit/combin/test_graph_facade.py:25 · validation/unit/combinatorics/test_kashiwara_crystal.py:32

numthy.MillerFunction.function

Return the function recorded by this MillerFunction.

(self) -> CurveFunction[E]

noether::numthy::MillerFunction · Python projection

Python projection convenience over this Rust mathematical type
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core
GIL attached / bounded
Semantic ID noether/numthy/MillerFunction@1

Executable usage: examples/16_rational_functions.py:3 · examples/26_elliptic_descent_functions.py:28 · examples/26_elliptic_descent_functions.py:29 · examples/26_elliptic_descent_functions.py:3 · examples/26_elliptic_descent_functions.py:31 · examples/26_elliptic_descent_functions.py:32

numthy.MillerFunction.multiplier

Return the multiplier recorded by this MillerFunction.

Detailed contract

Positive Miller multiplier n.

(self) -> int

noether::numthy::MillerFunction::multiplier · Direct Rust declaration

pub fn multiplier(&self) -> &Integer
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core
GIL attached / bounded
Semantic ID noether/numthy/MillerFunction@1

Executable usage: validation/unit/commalg/test_local_algebra.py:59 · validation/unit/commalg/test_polynomial_ideal_facade.py:328 · validation/unit/commalg/test_prime_localization.py:46 · validation/unit/commalg/test_prime_localization.py:47 · validation/unit/commalg/test_prime_localization.py:52 · validation/unit/homalg/test_local_module.py:34

numthy.MillerFunction.point

Return the point recorded by this MillerFunction.

Detailed contract

Source point P.

(self) -> EllipticCurvePoint[E]

noether::numthy::MillerFunction::point · Direct Rust declaration

pub fn point(&self) -> &EllipticCurvePoint<R>
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core
GIL attached / bounded
Semantic ID noether/numthy/MillerFunction@1

Executable usage: examples/26_elliptic_descent_functions.py:17 · examples/26_elliptic_descent_functions.py:18 · examples/26_elliptic_descent_functions.py:20 · examples/26_elliptic_descent_functions.py:25 · examples/26_elliptic_descent_functions.py:31 · examples/26_elliptic_descent_functions.py:32

numthy.MillerFunction.terminal_point

Return the terminal point recorded by this MillerFunction.

Detailed contract

Terminal group-law point nP.

(self) -> EllipticCurvePoint[E]

noether::numthy::MillerFunction::terminal_point · Direct Rust declaration

pub fn terminal_point(&self) -> &EllipticCurvePoint<R>
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core
GIL attached / bounded
Semantic ID noether/numthy/MillerFunction@1

Executable usage: validation/unit/numthy/test_elliptic_curve_facade.py:69

numthy.MillerFunction.to_json

Serialize this MillerFunction to its canonical typed JSON representation.

Detailed contract

Serialize source data; the function is recomputed on restoration.

(self) -> str

noether::numthy::MillerFunction::to_json · Direct Rust declaration

pub fn to_json(&self) -> Result<String, NoetherError>where R: Serialize, R::Element: Serialize,
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core
GIL attached / bounded
Semantic ID noether/numthy/MillerFunction@1

Executable usage: validation/unit/category/test_category.py:14 · validation/unit/category/test_category.py:15 · validation/unit/combin/test_acceptance_workload.py:30 · validation/unit/combin/test_acceptance_workload.py:65 · validation/unit/combin/test_acceptance_workload.py:86 · validation/unit/combin/test_graph_facade.py:106

numthy.MillerFunction.verify

Replay and verify the mathematical evidence carried by this MillerFunction.

Detailed contract

Recompute the complete double-and-add function and divisor formula.

(self) -> None

noether::numthy::MillerFunction::verify · Direct Rust declaration

pub fn verify(&self) -> Result<(), VerificationError>
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core
GIL attached / bounded
Semantic ID noether/numthy/MillerFunction@1

Executable usage: examples/01_bezout_certificates.py:10 · examples/01_bezout_certificates.py:3 · examples/03_exact_linear_systems.py:13 · examples/03_exact_linear_systems.py:3 · examples/04_smith_normal_form.py:12 · examples/06_shortest_lattice_vectors.py:12

numthy.MillerFunction.verify_principal_divisor

Compute verify principal divisor for this MillerFunction.

Detailed contract

Verify the divisor formula independently by complete factorization over Q. Verify the divisor formula independently by finite-field factorization.

(self) -> None

noether::numthy::MillerFunction::verify_principal_divisor · Direct Rust declaration

pub fn verify_principal_divisor(&self) -> Result<(), VerificationError>
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core
GIL attached / bounded
Semantic ID noether/numthy/MillerFunction@1

Executable usage: validation/unit/commalg/test_curve_facade.py:141 · validation/unit/commalg/test_curve_facade.py:63